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In mathematics, Mahler's theorem, introduced by , expresses continuous ''p''-adic functions in terms of polynomials. In any field, one has the following result. Let : be the forward difference operator. Then for polynomial functions ''f'' we have the Newton series: : where : is the ''k''th binomial coefficient polynomial. Over the field of real numbers, the assumption that the function ''f'' is a polynomial can be weakened, but it cannot be weakened all the way down to mere continuity. Mahler's theorem states that if ''f'' is a continuous p-adic-valued function on the ''p''-adic integers then the same identity holds. The relationship between the operator Δ and this polynomial sequence is much like that between differentiation and the sequence whose ''k''th term is ''x''''k''. It is remarkable that as weak an assumption as continuity is enough; by contrast, Newton series on the complex number field are far more tightly constrained, and require Carlson's theorem to hold. It is a fact of algebra that if ''f'' is a polynomial function with coefficients in any field of characteristic 0, the same identity holds where the sum has finitely many terms. ==References== * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mahler's theorem」の詳細全文を読む スポンサード リンク
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